Summary of the paper This paper presents a semi-parametric tuning-free procedure for estimating sparse concentration matrices. This method is applicable to the elliptical distribution family, while most of its competitors only apply to the sub-Gaussian distribution family. The procedure, called ALICE, learns the precision matrix column by column in a similar fashion than the CLIME (Cai et al, 2011), yet with important modifications: a first step is designed to learn the correlation matrix and the associated variances/standard deviations by means of the Kendall's Tau statistic as proposed in Liu et al, 2012. Then, the standard deviations are estimated through a recent proposal of Catoni (2012). In the second step, the inverse correlation is recovered by plugin-in the correlation estimated in the first step in a convex program similar to the CLIME, yet with a modification that allows for some calibration between the columns.

We propose a semiparametric method for estimating sparse precision matrix of high dimensional elliptical distribution. The proposed method calibrates regularizations when estimating each column of the precision matrix. Thus it not only is asymptotically tuning free, but also achieves an improved finite sample performance. Theoretically, we prove that the proposed method achieves the parametric rates of convergence in both parameter estimation and model selection. We present numerical results on both simulated and real datasets to support our theory and illustrate the effectiveness of the proposed estimator.

Interpretability of intelligent algorithms represented by deep learning has been yet an open problem. We discuss the shortcomings of the existing explainable method based on the two attributes of explanation, which are called completeness and explicitness. Furthermore, we point out that a model that completely relies on feed-forward mapping is extremely easy to cause inexplicability because it is hard to quantify the relationship between this mapping and the final model. Based on the perspective of the data space division, the principle of complete local interpretable model-agnostic explanations (CLIMEP) is proposed in this paper. To study the classification problems, we further discussed the equivalence of the CLIMEP and the decision boundary. As a matter of fact, it is also difficult to implementation of CLIMEP. To tackle the challenge, motivated by the fact that a fully-connected neural network (FCNN) with piece-wise linear activation functions (PWLs) can partition the input space into several linear regions, we extend this result to arbitrary FCNNs by the strategy of linearizing the activation functions. Applying this technique to solving classification problems, it is the first time that the complete decision boundary of FCNNs has been able to be obtained. Finally, we propose the DecisionNet (DNet), which divides the input space by the hyper-planes of the decision boundary. Hence, each linear interval of the DNet merely contains samples of the same label. Experiments show that the surprising model compression efficiency of the DNet with an arbitrary controlled precision.

At last, it is here! Few new technologies have ever been more anticipated and more predicted than the self-driving car. Anyone who drives cannot help but imagine not having to drive any more. It has been said that they will change our cities, our homes, our commerce, even our fundamental way of life. But at the same time, the actual progress has seemed … well … glacial, to the casual driver's eye.

At last, it is here! Few new technologies have ever been more anticipated and more predicted than the self-driving car. Anyone who drives cannot help but imagine not having to drive any more. It has been said that they will change our cities, our homes, our commerce, even our fundamental way of life. But at the same time, the actual progress has seemed … well … glacial, to the casual driver's eye.

Gaussian Graphical Models (GGMs) have wide-ranging applications in machine learning and the natural and social sciences. In most of the settings in which they are applied, the number of observed samples is much smaller than the dimension and they are assumed to be sparse. While there are a variety of algorithms (e.g. Graphical Lasso, CLIME) that provably recover the graph structure with a logarithmic number of samples, they assume various conditions that require the precision matrix to be in some sense well-conditioned. Here we give the first polynomial-time algorithms for learning attractive GGMs and walk-summable GGMs with a logarithmic number of samples without any such assumptions. In particular, our algorithms can tolerate strong dependencies among the variables. We complement our results with experiments showing that many existing algorithms fail even in some simple settings where there are long dependency chains, whereas ours do not.

This paper proposes a new method for estimating sparse precision matrices in the high dimensional setting. It has been popular to study fast computation and adaptive procedures for this problem. We propose a novel approach, called Sparse Column-wise Inverse Operator, to address these two issues. We analyze an adaptive procedure based on cross validation, and establish its convergence rate under the Frobenius norm. The convergence rates under other matrix norms are also established. This method also enjoys the advantage of fast computation for large-scale problems, via a coordinate descent algorithm. Numerical merits are illustrated using both simulated and real datasets. In particular, it performs favorably on an HIV brain tissue dataset and an ADHD resting-state fMRI dataset.

We propose a semiparametric procedure for estimating high dimensional sparse inverse covariance matrix. Our method, named ALICE, is applicable to the elliptical family. Computationally, we develop an efficient dual inexact iterative projection (${\rm D_2}$P) algorithm based on the alternating direction method of multipliers (ADMM). Theoretically, we prove that the ALICE estimator achieves the parametric rate of convergence in both parameter estimation and model selection. Moreover, ALICE calibrates regularizations when estimating each column of the inverse covariance matrix. So it not only is asymptotically tuning free, but also achieves an improved finite sample performance. We present numerical simulations to support our theory, and a real data example to illustrate the effectiveness of the proposed estimator.